Tensor ideals of abelian type and quantum groups

TL;DR

This paper explores tensor ideals in monoidal categories, focusing on quantum groups and finite group representations, establishing new connections with Duflo involutions and proving Lusztig's conjectures for Coxeter groups.

Contribution

It introduces methods to study tensor ideals in monoidal categories and applies them to quantum groups, connecting with Duflo involutions and proving key conjectures.

Findings

Established a framework for tensor ideals in monoidal categories.

Connected Duflo involutions with tensor ideals in quantum groups.

Proved Lusztig's conjectures for arbitrary Coxeter groups at equal parameters.

Abstract

We initiate a study of tensor ideals in linear rigid monoidal categories that are kernels of linear monoidal functors to abelian monoidal categories. We develop general methods and apply them to the category of tilting modules over quantum groups as well as to some representation categories of finite groups. In an appendix on Duflo involutions in monoidal categories, we make a connection between Duflo involutions in the affine Weyl group and tensor ideals for quantum groups, and prove some of Lusztig's conjectures for arbitrary Coxeter groups, at equal parameters, without invoking the boundedness hypothesis.